A generalized model of a resource-population system

Summary

A system of nonlinear differential equations describing a resource-population system is analyzed in terms of the existence and characteristics of its equilibrium states.

It is proved that, under the condition that k<αβ (necessary condition for the population being able to grow under optimal conditions), it is a necessary and sufficient condition for the system to have a steady state that the resource input rate to the system be constant.

When the resource input rate is a constant different from zero, the system has only one equilibrium point, at M ⁰=β/⁰/k, A ⁰=−(β/⁰/kα)ln(1−k/αβ), and this equilibrium point is always stable. In other words, the system population-resource will always reach the steady state, either monotonically (node) or by damped oscillations (focus), from any arbitrary initial condition in the positive quadrant.

When the resource input rate is equal to zero, the system has an infinite number of equilibrium points at M ₀=0, A ₀=constant. All these equilibrium points are unstable in the sense that any slight increase in M will move the system away from the equilibrium states, except for the point M ₀=0, A ₀=0, which is the only stable equilibrium point, to which the system will tend. This stable equilibrium point corresponds to the condition of complete annihilation of both resource and population.

Finally, it is proved that the system does not have limit cycles in the positive quadrant and is therefore incapable of self-oscillations.